THE TIME DEPENDENT SCHRÖDINGER EQUATION IN A STRONG LASER FIELD

Laser-matter interaction is one of the main topics in current physical research, due to the fact that lasers enable the study of matter under various physical conditions. For instance the interactions of lasers with plasmas, solid state matter as well as warm dense matter (WDM) are currently vivid research subjects. Laser interactions not only extended our understanding of matter itself, it also opens up possibilities for a wide range of applications. Cheaper and more effective cancer therapy and laser particle acceleration are promising outlooks [1, 2]. Lasers are also utilized to create WDM. For the production of WDM, laser shock waves are shot at a target, which will create highly compressed states of mega-bar pressures and temperatures on a scale of several electron volts [3, 4]. To probe WDM, lasers can be used for wide or small angle x-ray scattering analysis, also absorption spectroscopy and many other applications regarding this subject. Laser-driven plasma acceleration does not only hold a potential application to medical treatments, but in the area of material science, where lasers could facilitate a deeper implantation of multiple ion species in semiconductors [5]. In nuclear fusion research, lasers can compress the fusion pellet directly or indirectly. Ignition of the fusion reaction is then possible either due to the shock conditions or a additional external driver like electron or ion beams, which themselves are accelerated using powerful lasers [5]. A different, yet powerful, laser concept is the free-electron laser (FEL). The basic principle of FELs is the emittance of synchrotron radiation from relativistic electrons by sending them into an undulator. Thus, the electrons start to oscillate and emit light down to wavelengths on the x-ray spectrum [6]. FELs capable of emitting such wavelengths are called XFELs. Those kind of FELs are not only of interest in physical research, but also find applications in e.g. biological research [7]. Facilities featuring such XFELs are for instance the Lineac Coherent Light Source (LCLS) in Stanford [8] or, by the end of 2016, the European XFEL facility at DESY in Hamburg [9]. Usual laser concepts can be grouped, by the length of their emitted pulses, into ultra-short and long-pulse lasers. An example of an ultra-short pulse laser is the Dresden laser acceleration source (DRACO) at the Helmholtz Zentrum Dresden Rossendorf (HZDR) [10]. These kind of lasers are often deployed for laser particle acceleration research. Long-pulse lasers are used for the ignition of plasmas, as in the NIF facility. Other examples for long-pulse lasers are the n- and phelix [11, 12] at the GSI in Darmstadt or the Orion laser in Aldermaston [13]. The scale of those long laser pulses is about several nanoseconds, whereas short-pulse lasers have typical pulse lengths of several femtoseconds. The physical effects of laser matter interaction include the creation of Auger electrons, where a laser pulse pulls out an inner shell electron and causes the relaxation of an outer shell electron to the unoccupied energy level. This relaxation causes the emission of another electron, the so called Auger electron, without the emittance of any radiation [14]. The Auger electron effect already shows that lasers are able to ionize atoms, where an electron essentially becomes free through the interaction with a laser field. This can lead to further effects like the recombination of electrons with ions, which can cause the emission of photons as the so called radiative recombination. Lasers also are able to interact with free electrons by inelastic scattering. Electrons can get excited to higher energy levels by inelastic laser scattering, afterwards this excited state relaxes into its initial energy level by the emission of a photon [14]. In addition to all those bound electron laser interactions as examples of inelastic scattering, elastic scattering of bound electrons is possible as well. A large part of the x-ray scattering spectrum is the elastic peak, which is caused by elastic interactions of bound electrons with x-ray wavelength photons. This means, 5that bound electrons oscillate in the laser field and emit light of the same frequency. That is the case, if the photon energy does not match the transition energy. In general, laser emissions are able to lift bound states to higher energy levels. Furthermore lasers enable lowering of potential barriers, enabling a bound particle to tunnel trough [15]. All of those effects are experimentally observed and broaden our knowledge about matter. Measurements of these physical effects are standard nowadays. However, it is very challenging to temporally resolve the processes, as they happen on atto- to femto-second time scales. Simulating those physical processes enables us to see the fundamental process behind experimental observations, and enhance the way theory and experiment complement each other. Especially, effects like ionization or state transitions are only well understood, if one is able to exactly explain what is happening and how well the theory behind those processes predict the experimental outcome on those microscopic scales . As one tries to explain a non relativistic quantum mechanical system, the starting point of all models is the Schrödinger Equation. Thus, the main subject of this thesis is to find a general way to simulate a one-dimensional quantum system in a strong time-dependent potential